Variable Range Hopping Conduction in Complex Systems and a Percolation Model with Tunneling

For the low-temperature electrical conductance of a disordered {\it quantum insulator} in $d$-dimensions, Mott \cite{mott} had proposed his Variable Range Hopping (VRH) formula, $G(T) = G_0 {\rm exp}[-(T_0/T)^{\gamma}]$, where $G_0$ is a material constant and $T_0$ is a characteristic temperature scale. For disordered but non-interacting carrier charges, Mott had found that $\gamma = 1/(d+1)$ in $d$-dimensions. Later on, Efros and Shkolvskii \cite{esh} found that for a pure ({\it i.e.}, disorder-free) {\it quantum insulator} with interacting charges, $\gamma =1/2$, {\it independent of d}. Recent experiments indicate that $\gamma$ is either (i) larger than any of the above predictions; and, (ii) more intriguingly, it seems to be a function of $p$, the dopant concentration. We investigate this issue with a {\it semi-classical} or {\it semi-quantum} RRTN ({\it Random Resistor cum Tunneling-bond Network}) model, developed by us in the 1990's. These macroscopic {\it granular/ percolative composites} are built up from randomly placed meso- or nanoscopic coarse-grained clusters, with two phenomenological functions for the temperature-dependence of the metallic and the semi-conducting bonds. We find that our RRTN model (in 2D, for simplicity) also captures this continuous change of $\gamma$ with $p$, satisfactorily.